# Properties

 Label 1600r Number of curves 4 Conductor 1600 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1600.w1")

sage: E.isogeny_class()

## Elliptic curves in class 1600r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1600.w3 1600r1 [0, -1, 0, -133, -363]  384 $$\Gamma_0(N)$$-optimal
1600.w4 1600r2 [0, -1, 0, 367, -2863]  768
1600.w1 1600r3 [0, -1, 0, -4133, 103637]  1152
1600.w2 1600r4 [0, -1, 0, -3633, 129137]  2304

## Rank

sage: E.rank()

The elliptic curves in class 1600r have rank $$0$$.

## Modular form1600.2.a.w

sage: E.q_eigenform(10)

$$q + 2q^{3} + 2q^{7} + q^{9} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels. 